@Ashvin Yes you are correct that there is this technical assumption concerning isolated points in the intersection of the tropical intersections, which is not as satisfactory. The reference Michael Stoll discusses the stable tropical intersection, which is a well-defined number. If I recall, the issue with this approach and your question is that it is not clear how the stable tropical intersection is related to "natural" invariants of $f$ and $g$.

I believe the Joe Rabinoff has some work on this topic e.g., arxiv.org/abs/1007.2665. Also, you might be able to find some useful things in Section 5 of arxiv.org/pdf/1708.07057.pdf. I believe that in order to get bounds in terms of the Newton polytopes you need to understand a bit about the tropicalization of the common zeros.

Have you checked out Sections 3 and 4 of arxiv.org/pdf/2105.13587.pdf. The authors define a theory of adelic line bundles and integration pairings using Berkovich spaces over $\mathbb{Z}$ and relate this to classical Arakelov intersection pairings.

Perhaps you can find the answer to your question in Section 3 of arxiv.org/pdf/1410.2216.pdf? E.g., Corollary 3.6 of that work seems very relevant.

For your second question, the invariant $a'(D)$ is the upper limit of the integral in asymptotic volume constants $\beta(-K_X,D)$ (see e.g., the definition of $\gamma_{\text{eff}}$ in this paper of McKinnon--Roth). In more generality, one defines $$\beta(-K_X,D) = \int_0^{\infty} \frac{\text{vol}(-K_X - tD)}{\text{vol}(-K_X)} dt.$$ One can show that the upper limit of this integral is $a'(D)$ as you have defined it. If you would like more references on asymptotic volume constants (or beta constants), please let me know.

If you want to see how this is related to a notion of “integration” you should look up Colmez integration.